0.3 Discrete structures logic  (Page 8/23)

To prove this, first let us get rid of → using one of the identities: (P→Q ) ⇔( ¬P ⋁Q).

That is, ¬( P →Q ) ⇔¬( ¬P ⋁Q ).

Then by De Morgan, it is equivalent to ¬¬P ⋀¬Q , which is equivalent to P ⋀¬Q, since the double negation of a proposition is equivalent to the original proposition as seen in the identities.

2. P ⋁( P ⋀Q ) ⇔P --- Absorption

What this tells us is that P ⋁( P ⋀Q ) can be simplified to P, or if necessary P can be expanded into P ⋁( P ⋀Q ) .

To prove this, first note that P ⇔( P ⋀T ).

Hence

P ⋁( P ⋀Q )

⇔( P ⋀T ) ⋁( P ⋀Q )

⇔P ⋀( T ⋁Q ) , by the distributive law.

⇔( P ⋀T ) , since ( T ⋁Q ) ⇔T.

⇔P , since ( P ⋀T ) ⇔P.

Note that by the duality

P ⋀( P ⋁Q ) ⇔P also holds.

Implications

The following implications are some of the relationships between propositions that can be derived from the definitions (meaning) of connectives. ⇒ below corresponds to → and it means that the implication always holds. That is it is a tautology.

These implications are used in logical reasoning. When the right hand side of these implications is substituted for the left hand side appearing in a proposition, the resulting proposition is implied by the original proposition, that is, one can deduce the new proposition from the original one.

First the implications are listed, then examples to illustrate them are given. List of Implications:

1. P ⇒(P ⋁Q) ----- addition

2. (P ⋀Q) ⇒P ----- simplification

3. [P ⋀(P →Q] ⇒Q ----- modus ponens

4. [(P →Q) ⋀¬Q] ⇒¬P ----- modus tollens

5. [ ¬P ⋀(P ⋁Q] ⇒Q ----- disjunctive syllogism

6. [(P →Q) ⋀(Q→R)] ⇒(P→R) ----- hypothetical syllogism

7. (P→Q) ⇒[(Q→R)→(P→R)]

8. [(P→Q) ⋀(R→S)] ⇒[(P ⋀R)→(Q ⋀S)]

9. [(P ↔Q) ⋀(Q ↔R)] ⇒(P ↔R)

Examples:

1. P ⇒(P ⋁Q) ----- addition

For example, if the sun is shining, then certainly the sun is shining or it is snowing. Thus

"if the sun is shining, then the sun is shining or it is snowing." "If 0<1, then 0 ≤1 or a similar statement is also often seen.

2. (P ⋀Q) ⇒P ----- simplification

For example, if it is freezing and (it is) snowing, then certainly it is freezing. Thus "If it is freezing and (it is) snowing, then it is freezing."

3. [P ⋀(P →Q] ⇒Q ----- modus ponens

For example, if the statement "If it snows, the schools are closed" is true and it actually snows, then the schools are closed.

This implication is the basis of all reasoning. Theoretically, this is all that is necessary for reasoning. But reasoning using only this becomes very tedious.

4. [(P →Q) ⋀¬Q] ⇒¬P ----- modus tollens

For example, if the statement "If it snows, the schools are closed" is true and the schools are not closed, then one can conclude that it is not snowing. Note that this can also be looked at as the application of the contrapositive and modus ponens. That is, (P→Q) is equivalent to ( ¬Q )→( ¬P ). Thus if in addition ¬Q holds, then by the modus ponens, ¬P is concluded.

5. [ ¬P ⋀(P ⋁Q] ⇒Q ----- disjunctive syllogism

For example, if the statement "It snows or (it) rains." is true and it does not snow, then one can conclude that it rains.

6. [(P→Q) ⋀(Q→R)] ⇒(P→R) ----- hypothetical syllogism